The shortest and simplest answer I can give is about something called Rayleigh scattering. As light comes through the atmosphere, a lot of colors pass right through. Blue doesn't, though. It gets absorbed and radiated all over the place so that, everywhere you look in the atmosphere, you see that radiated blue light. The result is the sky appearing blue.

1)
The short answer:

It is because blue light from the sun strikes the air molecules and scatters and our eyes perceive it as blue.

2)
Watch this video:
http://www.youtube.com/watch?v=u4hw-aXiQmc

In 3 minutes, this video will explain to you exactly why the sky is blue and will also show you illustrations that will help you completely understand it. This video also goes above-and-beyond to provide you with the A+ answer explaining why sunsets are orange!

If for some reason the video reports that it is "no longer available", simply click in the video window to be taken to the video directly on YouTube. It will play from there.

Paragraph by paragraph, this site also gives you a very nice explanation:
http://www.ehow.com/how-does_4574391_why-sky-blue.html

"A clear cloudless day-time sky is blue because molecules in the air scatter blue light from the sun more than they scatter red light. (However;) (w)hen we look towards the sun at sunset, we see red and orange colours because the blue light has been scattered out and away from the line of sight." Original by Philip Gibbs May 1997.

The blue color of the sky is due to Rayleigh scattering. As light moves through the atmosphere, most of the longer wavelengths pass straight through. Little of the red, orange and yellow light is affected by the air.

However, much of the shorter wavelength light is absorbed by the gas molecules. The absorbed blue light is then radiated in different directions. It gets scattered all around the sky. Whichever direction you look, some of this scattered blue light reaches you. Since you see the blue light from everywhere overhead, the sky looks blue.

As you look closer to the horizon, the sky appears much paler in color. To reach you, the scattered blue light must pass through more air. Some of it gets scattered away again in other directions. Less blue light reaches your eyes. The color of the sky near the horizon appears paler or white.

Why not violet?
If shorter wavelengths are scattered most strongly, then there is a puzzle as to why the sky does not appear violet, the colour with the shortest visible wavelength. The spectrum of light emission from the sun is not constant at all wavelengths, and additionally is absorbed by the high atmosphere, so there is less violet in the light. Our eyes are also less sensitive to violet. That's part of the answer; yet a rainbow shows that there remains a significant amount of visible light coloured indigo and violet beyond the blue. The rest of the answer to this puzzle lies in the way our vision works. We have three types of colour receptors, or cones, in our retina. They are called red, blue and green because they respond most strongly to light at those wavelengths. As they are stimulated in different proportions, our visual system constructs the colours we see.

Basically, blue has the shortest wave length and scattered. So, during the day, the sky is blue because molecules in the air scatter blue light from the sun more than they scatter red light. When our eyes see the sun at sunset, we are viewing red and orange colors because the blue light has been scattered out and away from the line of sight. During the night time, the sky sometimes looks orange or red because of air pollution. Dust and other floating particles in the air act as a filter on the sunlight. When the sun is low the air layer is thicker and the light is more filtered, so it looks yellow, orange and finally red.

A simpler way of explaining it is that during the day, when the sun is out, the light from the sun reflects off the ocean (which is mainly a blue color) and transfers the color onto the sky, which is why we say the sky is blue. For example, if people decided that they should dye the sea purple, the results would be that the sky would change to the same color.

At night, the sun sets and it stops reflecting off the ocean, which is why at night we do not have a blue sky.

In scientific detail:

Light of a particular color is determined by its frequency. The higher the frequency, the shorter the wavelength, which means the more blue it appears.

The sun's light is made up of many frequencies that when mixed together produce white light. Rainbows are created by diffraction with precipitation of rain in or water in the air which changes the directions of intensities of a group of waves, making the sky appear to be multi-colorful across the spectrum, splitting this light into several frequencies. Our atmosphere on earth is filled with plenty of dust particles that act like a filter, scattering the light rays. Light rays with longer wavelengths such as reds and yellows will usually travel more easily through the earth's atmosphere, while the rays with the shorter wavelengths, like blues and indigos, will usually be dispersed more easily. Therefore the light rays that disperse more easily give the sky the color blue.

The sky is not always blue:

Have you noticed red skies at sunrise or sunset? These skies work with the same phenomenon. Light hits our planet Earth at an angle and has more of the atmosphere to go through, which causes a filtering effect that makes our sky red.

The sky is blue because light is scattered when it hits the atmosphere. As light passes through the waves of shorter wavelength are scattered by molecules in the atmosphere, but longer wavelength live can pass through. Blue, being on the shorter wavelength side of the spectrum is scattered by molecules in the air and so we are able to see it. However, as the angle at which the sun's light is hitting a certain part of the Earth's atmosphere changes (as in sunset), more colors in the spectrum are scattered in the atmosphere, generating more color in the sky. This also happens at sunrise, though it is happening in reverse. This process of the scattering of light, also accounts for the grayish look of the sky on a overcast day. When it is overcast, more light is being scattered due to the clouds, and therefore the overall color is a mix of all the colors of the spectrum.

http://static.howstuffworks.com/gif/question-39-1.jpg

The blue color of the sky is due to Rayleigh scattering. As light moves through the atmosphere, most of the longer wavelengths pass straight through. Little of the red, orange and yellow light is affected by the air.

However, much of the shorter wavelength light is absorbed by the gas molecules. The absorbed blue light is then radiated in different directions. It gets scattered all around the sky. Whichever direction you look, some of this scattered blue light reaches you. Since you see the blue light from everywhere overhead, the sky looks blue.

http://www.sciencemadesimple.com/img_sky/bluesky.gif

As you look closer to the horizon, the sky appears much paler in color. To reach you, the scattered blue light must pass through more air. Some of it gets scattered away again in other directions. Less blue light reaches your eyes. The color of the sky near the horizon appears paler or white.

http://www.sciencemadesimple.com/img_sky/horiz.gif

For more info:
http://spaceplace.nasa.gov/en/kids/misrsky/misr_sky.shtml
http://science.howstuffworks.com/sky.htm
http://www.enchantedlearning.com/subjects/astronomy/planets/earth/Skyblue.shtml
http://www.sky-watch.com/articles/skyblue.html
http://world.std.com/~mmcirvin/bluesky.html